Powers of Elements in Group Direct Product

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Theorem

Let $\struct {G, \circ_1}$ and $\struct {H, \circ_2}$ be group whose identities are $e_G$ and $e_H$.

Let $\struct {G \times H, \circ}$ be the group direct product of $G$ and $H$.


Then:

$\forall n \in \Z: \forall g \in G, h \in H: \tuple {g, h}^n = \tuple {g^n, h^n}$


Proof

Proof by induction:

For all $n \in \N$, let $\map P n$ be the proposition $\forall g \in G, h \in H: \tuple {g, h}^n = \tuple {g^n, h^n}$.


Basis for the Induction

$\map P 0$ is true, as this says:

$\tuple {g, h}^0 = \tuple {e_G, e_H}$

$\map P 1$ is true, as this says:

$\tuple {g, h} = \tuple {g, h}$

This is our basis for the induction.


Induction Hypothesis

Now we need to show that, if $\map P k$ is true, where $k \ge 1$, then it logically follows that $\map P {k + 1}$ is true.


So this is our induction hypothesis:

$\tuple {g, h}^k = \tuple {g^k, h^k}$


Then we need to show:

$\tuple {g, h}^{k + 1} = \tuple {g^{k + 1}, h^{k + 1} }$


Induction Step

This is our induction step:

\(\ds \tuple {g, h}^{k + 1}\) \(=\) \(\ds \tuple {g, h}^k \circ \tuple {g, h}\)
\(\ds \) \(=\) \(\ds \tuple {g^k, h^k} \circ \tuple {g, h}\) Induction Hypothesis
\(\ds \) \(=\) \(\ds \tuple {g^k \circ_1 g, h^k \circ_2 h}\) Definition of Group Direct Product
\(\ds \) \(=\) \(\ds \tuple {g^{k + 1}, h^{k + 1} }\)


So $\map P k \implies \map P {k + 1}$ and the result follows by the Principle of Mathematical Induction.


Therefore:

$\forall n \in \N: \forall g \in G, h \in H: \tuple {g, h}^n = \tuple {g^n, h^n}$


So we have shown the result holds true for all $n \ge 0$.

The result for $n < 0$ follows directly from Powers of Group Elements for Negative Indices.

$\blacksquare$